Resilience of neural networks for locomotion.

Locomotion is an essential behaviour for the survival of all animals. The neural circuitry underlying locomotion is therefore highly robust to a wide variety of perturbations, including injury and abrupt changes in the environment. In the short term, fault tolerance in neural networks allows locomotion to persist immediately after mild to moderate injury. In the longer term, in many invertebrates and vertebrates, neural reorganization including anatomical regeneration can restore locomotion after severe perturbations that initially caused paralysis. Despite decades of research, very little is known about the mechanisms underlying locomotor resilience at the level of the underlying neural circuits and coordination of central pattern generators (CPGs). Undulatory locomotion is an ideal behaviour for exploring principles of circuit organization, neural control and resilience of locomotion, offering a number of unique advantages including experimental accessibility and modelling tractability. In comparing three well-characterized undulatory swimmers, lampreys, larval zebrafish and Caenorhabditis elegans, we find similarities in the manifestation of locomotor resilience. To advance our understanding, we propose a comparative approach, integrating experimental and modelling studies, that will allow the field to begin identifying shared and distinct solutions for overcoming perturbations to persist in orchestrating this essential behaviour.

The role of curvature feedback in the energetics and dynamics of lamprey swimming: A closed-loop model.

Like other animals, lampreys have a central pattern generator (CPG) circuit that activates muscles for locomotion and also adjusts the activity to respond to sensory inputs from the environment. Such a feedback system is crucial for responding appropriately to unexpected perturbations, but it is also active during normal unperturbed steady swimming and influences the baseline swimming pattern. In this study, we investigate different functional forms of body curvature-based sensory feedback and evaluate their effects on steady swimming energetics and kinematics, since little is known experimentally about the functional form of curvature feedback. The distributed CPG is modeled as chains of coupled oscillators. Pairs of phase oscillators represent the left and right sides of segments along the lamprey body. These activate muscles that flex the body and move the lamprey through a fluid environment, which is simulated using a full Navier-Stokes model. The emergent curvature of the body then serves as an input to the CPG oscillators, closing the loop. We consider two forms of feedback, each consistent with experimental results on lamprey proprioceptive sensory receptors. The first, referred to as directional feedback, excites or inhibits the oscillators on the same side, depending on the sign of a chosen gain parameter, and has the opposite effect on oscillators on the opposite side. We find that directional feedback does not affect beat frequency, but does change the duration of muscle activity. The second feedback model, referred to as magnitude feedback, provides a symmetric excitatory or inhibitory effect to oscillators on both sides. This model tends to increase beat frequency and reduces the energetic cost to the lamprey when the gain is high and positive. With both types of feedback, the body curvature has a similar magnitude. Thus, these results indicate that the same magnitude of curvature-based feedback on the CPG with different functional forms can cause distinct differences in swimming performance.

Mixing and pumping by pairs of helices in a viscous fluid.

Here, we study the fluid dynamics of a pair of rigid helices rotating at a constant velocity, tethered at their bases, in a viscous fluid. Our computations use a regularized Stokeslet framework, both with and without a bounding plane, so we are able to discern precisely what flow features are unaccounted for in studies that ignore the surface from which the helices emanate. We examine how the spacing and phase difference between identical rotating helices affects their pumping ability, axial thrust, and power requirements. We also find that optimal mixing of the fluid around two helices is achieved when they rotate in opposite phase, and that the mixing is enhanced as the distance between the helices decreases.

The effect of intrinsic muscular nonlinearities on the energetics of locomotion in a computational model of an anguilliform swimmer.

Animals move through their environments using muscles to produce force. When an animal׳s nervous system activates a muscle, the muscle produces different amounts of force depending on its length, its shortening velocity, and its time history of force production. These muscle forces interact with forces from passive tissue properties and forces from the external environment. Using an integrative computational model that couples an elastic, actuated model of an anguilliform, lamprey-like swimmer with a surrounding Navier-Stokes fluid, we study the effects of this coupling between the muscle force and the body motion. Swimmers with different forms of this coupling can achieve similar motions, but use different amounts of energy. The velocity dependence is the most important property of the ones we considered for reducing energy costs and helping us to stabilize oscillations. These effects are strongly influenced by how rapidly the muscle deactivates; if force decays too slowly, muscles on opposite sides of the body end up fighting each other, increasing energy cost. Work-dependent deactivation, an effect that causes a muscle to deactivate more rapidly if it has recently produced mechanical work, works together with the velocity dependence to reduce the energy cost of swimming.

The role of mechanical resonance in the neural control of swimming in fishes.

The bodies of many fishes are flexible, elastic structures; if you bend them, they spring back. Therefore, they should have a resonant frequency: a bending frequency at which the output amplitude is maximized for a particular input. Previous groups have hypothesized that swimming at this resonant frequency could maximize efficiency, and that a neural circuit called the central pattern generator might be able to entrain to a mechanical resonance. However, fishes swim in water, which may potentially damp out many resonant effects. Additionally, their bodies are elongated, which means that bending can occur in complicated ways along the length of the body. We review previous studies of the mechanical properties of fish bodies, and then present new data that demonstrate complex bending properties of elongated fish bodies. Resonant peaks in amplitude exist, but there may be many of them depending on the body wavelength. Additionally, they may not correspond to the maximum swimming speed. Next, we describe experiments using a closed-loop preparation of the lamprey, in which a preparation of the spinal cord is linked to a real-time simulation of the muscle and body properties, allowing us to examine resonance entrainment as we vary the simulated resonant frequency. We find that resonance entrainment does occur, but is rare. Gain had a significant, though weak, effect, and a nonlinear muscle model produced resonance entrainment more often than a linear filter. We speculate that resonance may not be a critical effect for efficient swimming in elongate, anguilliform swimmers, though it may be more important for stiffer carangiform and thunniform fishes.

Coupling biochemistry and hydrodynamics captures hyperactivated sperm motility in a simple flagellar model.

Hyperactivation in mammalian sperm is characterized by highly asymmetrical waveforms and an increase in the amplitude of flagellar bends. It is important for the sperm to be able to achieve hyperactivated motility in order to reach and fertilize the egg. Calcium (Ca(2+)) dynamics are known to play a large role in the initiation and maintenance of hyperactivated motility. Here we present an integrative model that couples the CatSper channel mediated Ca(2+) dynamics of hyperactivation to a mechanical model of an idealized sperm flagellum in a 3-d viscous, incompressible fluid. The mechanical forces are due to passive stiffness properties and active bending moments that are a function of the local Ca(2+) concentration along the length of the flagellum. By including an asymmetry in bending moments to reflect an asymmetry in the axoneme's response to Ca(2+), we capture the transition from activated motility to hyperactivated motility. We examine the effects of elastic properties of the flagellum and the Ca(2+) dynamics on the overall swimming patterns. The swimming velocities of the model flagellum compare well with data for hyperactivated mouse sperm.

Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming.

Animal movements result from a complex balance of many different forces. Muscles produce force to move the body; the body has inertial, elastic, and damping properties that may aid or oppose the muscle force; and the environment produces reaction forces back on the body. The actual motion is an emergent property of these interactions. To examine the roles of body stiffness, muscle activation, and fluid environment for swimming animals, a computational model of a lamprey was developed. The model uses an immersed boundary framework that fully couples the Navier-Stokes equations of fluid dynamics with an actuated, elastic body model. This is the first model at a Reynolds number appropriate for a swimming fish that captures the complete fluid-structure interaction, in which the body deforms according to both internal muscular forces and external fluid forces. Results indicate that identical muscle activation patterns can produce different kinematics depending on body stiffness, and the optimal value of stiffness for maximum acceleration is different from that for maximum steady swimming speed. Additionally, negative muscle work, observed in many fishes, emerges at higher tail beat frequencies without sensory input and may contribute to energy efficiency. Swimming fishes that can tune their body stiffness by appropriately timed muscle contractions may therefore be able to optimize the passive dynamics of their bodies to maximize peak acceleration or swimming speed.

A model of CatSper channel mediated calcium dynamics in mammalian spermatozoa.

CatSpers are calcium (Ca(2+)) channels that are located along the principal piece of mammalian sperm flagella and are directly linked to sperm motility and hyperactivation. It has been observed that Ca(2+) entry through CatSper channels triggers a tail to head Ca(2+) propagation in mouse sperm, as well as a sustained increase of Ca(2+) in the head. Here, we develop a mathematical model to investigate this propagation and sustained increase in the head. A 1-d reaction-diffusion model tracking intracellular Ca(2+) with flux terms for the CatSper channels, a leak flux, and plasma membrane Ca(2+) clearance mechanism is studied. Results of this simple model exhibit tail to head Ca(2+) propagation, but no sustained increase in the head. Therefore, in this model, a simple plasma membrane pump-leak system with diffusion in the cytosol cannot account for these experimentally observed results. It has been proposed that Ca(2+) influx from the CatSper channels induce additional Ca(2+) release from an internal store. We test this hypothesis by examining the possible role of Ca(2+) release from the redundant nuclear envelope (RNE), an inositol 1,4,5-trisphosphate (IP(3)) gated Ca(2+) store in the neck. The simple model is extended to include an equation for IP(3) synthesis, degradation, and diffusion, as well as flux terms for Ca(2+) in the RNE. When IP(3) and the RNE are accounted for, the results of the model exhibit a tail to head Ca(2+) propagation as well as a sustained increase of Ca(2+) in the head.

EVALUATION OF INTERFACIAL FLUID DYNAMICAL STRESSES USING THE IMMERSED BOUNDARY METHOD.

The goal of this paper is to examine the evaluation of interfacial stresses using a standard, finite difference based, immersed boundary method (IMBM). This calculation is not trivial for two fundamental reasons. First, the immersed boundary is represented by a localized boundary force which is distributed to the underlying fluid grid by a discretized delta function. Second, this discretized delta function is used to impose a spatially averaged no-slip condition at the immersed boundary. These approximations can cause errors in interpolating stresses near the immersed boundary.To identify suitable methods for evaluating stresses, we investigate three model flow problems at very low Reynolds numbers. We compare the results of the immersed boundary calculations to those achieved by the boundary element method (BEM). The stress on an immersed boundary may be calculated either by direct evaluation of the fluid stress (FS) tensor or, for the stress jump, by direct evaluation of the locally distributed boundary force (wall stress or WS). Our first model problem is Poiseuille channel flow. Using an analytical solution of the immersed boundary formulation in this simple case, we demonstrate that FS calculations should be evaluated at a distance of approximately one grid spacing inward from the immersed boundary. For a curved immersed boundary we present a procedure for selecting representative interfacial fluid stresses using the concepts from the Poiseuille flow test problem. For the final two model problems, steady state flow over a bump in a channel and unsteady peristaltic pumping, we present an 'exclusion filtering' technique for accurately measuring stresses. Using this technique, these studies show that the immersed boundary method can provide reliable approximations to interfacial stresses.

An integrative computational model of multiciliary beating.

The coordinated beating of motile cilia is responsible for ovum transport in the oviduct, transport of mucus in the respiratory tract, and is the basis of motility in many single-celled organisms. The beating of a single motile cilium is achieved by the ATP-driven activation cycles of thousands of dynein molecular motors that cause neighboring microtubule doublets within the ciliary axoneme to slide relative to each other. The precise nature of the spatial and temporal coordination of these individual motors is still not completely understood. The emergent geometry and dynamics of ciliary beating is a consequence of the coupling of these internal force-generating motors, the passive elastic properties of the axonemal structure, and the external viscous, incompressible fluid. Here, we extend our integrative model of a single cilium that couples internal force generation with the surrounding fluid to the investigation of multiciliary interaction. This computational model allows us to predict the geometry of beating, along with the detailed description of the time-dependent flow field both near and away from the cilia. We show that synchrony and metachrony can, indeed, arise from hydrodynamic coupling. We also investigate the effects of viscosity and neighboring cilia on ciliary beat frequency. Moreover, since we have precise flow information, we also measure the dependence of the total flow pumped per cilium per beat upon parameters such as viscosity and ciliary spacing.

Fluid dynamic models of flagellar and ciliary beating.

We have developed a fluid-mechanical model of a eucaryotic axoneme that couples the internal force generation of dynein molecular motors, the passive elastic mechanics of microtubules, and forces due to nexin links with a surrounding incompressible fluid. This model has been used to examine both ciliary beating and flagellar motility. In this article, we show preliminary simulation results for sperm motility in both viscous and viscoelastic fluids, as well as multiciliary interaction with a mucus layer.

A computational model of the mechanics of growth of the villous trophoblast bilayer.

We present a computational model of the mechanics of growth of the trophoblast bilayer in a chorionic villous, the basic structure of the placenta. The placental trophoblast is modeled as a collection of elastic neutrally buoyant membranes (mononuclear cytotrophoblasts and multinucleated syncytiotrophoblast) filled with a viscous, incompressible fluid (cytoplasm) with sources of growth located inside cells. We show how this complex, dynamic fluid-based structure can be modeled successfully using the immersed boundary method. The results of our research presented here include simulations of two processes-cell proliferation and cell fusion which both play a crucial role in the growth and development of the trophoblast tissue. We present the computed results of simulations of both processes running independently as well as simultaneously, along with comparisons with clinically obtained results.